Survey of Mathematical Models of and Numerical Methods for Fluid Dynamics Water Engineering

Abstract: Computational fluid dynamics (CFD) has become a cornerstone of modern water engineering, providing quantitative tools for the analysis, prediction, and management of complex hydraulic systems across a wide range of spatial and temporal scales. This survey reviews the mathematical models and numerical methods that underpin CFD applications in water engineering, from depth-averaged formulations such as the shallow water equations to fully three-dimensional Navier-Stokes models, as well as selected alternative modeling approaches. We examine the historical development of these models, their mathematical structure, and the numerical discretization and solution strategies commonly employed in practice, including finite difference, finite volume, and finite element methods. Beyond core solver technology, the survey addresses practical modeling issues such as source-term treatment, wetting and drying, turbulence modeling, free-surface representation, and computational efficiency. The growing role of data integration is also discussed, encompassing data assimilation, uncertainty quantification, and emerging machine-learning-assisted approaches that complement physics-based solvers. To illustrate the impact of modeling and numerical choices on real-world applications, representative case studies from large-scale water management systems are reviewed. By integrating theory, numerical techniques, and applied perspectives, this survey provides a unified reference for researchers and practitioners seeking to understand both the foundational principles and contemporary challenges of CFD in water engineering.